Uniform spaces and measures
The well-known Riesz representation theorem tells us that every linear functional on a space of the form \(C(X)\), where \(X\)~is compact Hausdorff, can be represented as an integral with respect to some measure on~\(X\). There have been various attempts at generalisations of this theorem and the volume under review provides a unified treatment in the class of uniform spaces. As the Lebesgue integral shows, when restricted to the space of uniformly continuous functions on the rationals, there will be functionals without representing measures. The main part of the book investigates when representing measures exist for functionals on various (sub)spaces of uniformly continuous functions. The central notion is that of a uniform measure. To define these one takes the pseudometric viewpoint of uniform spaces and considers for every compatible pseudometric, \(d\), the subspace \(U_d(X)\) of the space of uniformly continuous functions that consists of the functions that are \(1\)-Lipschitz continuous and have \(\ell_\infty\)-norm at most~\(1\). A \textit{uniform measure} is a functional on the space of uniformly continuous functions whose restriction to every~\(U_d(X)\) is continuous with respect to the topology of pointwise convergence. The book provides a solid base for further study and application of these measures, when they are induced by measures, when measures induce uniform measures, the structure of the space of uniform measures and related functionals. Well worth the effort although the reader should be warned that the author introduces quite a large amount of notation; this makes for succinct statements of the results but does make it difficult to browse for the result that one needs.
scientific article; zbMATH DE number 6062329